Oscillatory retarded functional systems

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www.elsevier.com/locate/jmaa

Oscillatory retarded functional systems

José M. Ferreira

a,∗,1

_{and Sandra Pinelas}

b

a_{Instituto Superior Técnico, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa, Portugal} b_{Universidade dos Açores, Departamento de Matemática, R. Mãe de Deus, 9500-321 Ponta Delgada, Portugal}

Received 24 December 2002 Submitted by S.R. Grace

Abstract

This note is concerned with the oscillatory behavior of a linear retarded system. Several criteria are obtained for having the system oscillatory. Conditions regarding the existence of nonoscillatory solutions are also given.

1. Introduction

The oscillation theory of delay equations has received a large amount of attention dur-ing the last two decades, as one can see through the textbooks [1–5] and references therein. However, excepting discrete difference systems and particular results which can be ob-tained through some studies regarding differential systems of neutral type, some gaps can be found in the literature with respect to functional retarded systems.

The aim of this note is to study the oscillatory behavior of the system

x(t)= 0 −1 xt− r(θ)dν(θ ), (1) *_{Corresponding author.}

E-mail address: [email protected] (J.M. Ferreira). 1_{Supported in part by FCT (Portugal).}

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where x(t)∈ Rn, r(θ ) is a real continuous function on[−1, 0], positive on [−1, 0[, and ν(θ ) is a real n-by-n matrix valued function of bounded variation on[−1, 0], which in case

of having r(0)= 0 will be assumed atomic at zero, that is, such that

lim γ→0+ 0 −γ dη(θ ) =0,

where for a given norm, · , in the space Mn(R), of all real n-by-n matrices, by b

a

dη(θ )

we mean the total variation of ν on an interval[a, b] ⊂ [−1, 0]. Notice that when r(θ) is positive on[−1, 0], no atomicity assumption on the function ν is necessary.

The system (1) for r(θ )= −rθ (r > 0) and θ ∈ [−1, 0], is the class of linear retarded functional systems x(t)= 0 −r x(t+ θ) dη(θ ), (2)

where η(θ )= ν(θ/r) is assumed to be atomic at zero. This is the most common general linear retarded functional system appearing in the literature (see [6]). Our preference on system (1) regards the possibility of understanding more clearly the role of the delays on the oscillatory behavior of functional retarded systems.

Considering the value r = max{r(θ): −1 θ 0}, a continuous function x : [− r ,

+∞[ → R, is said a solution of (1) if satisfies this equation for every t 0. A

solu-tion of (1), x(t)= [x1(t), . . . , xn(t)]T, is called oscillatory if every component, xj(t),

j = 1, . . ., n, has arbitrary large zeros; otherwise x(t) is said a nonoscillatory solution.

Whenever all solutions of (1) are oscillatory we will say that (1) is an oscillatory system. Both systems (1) and (2) include the important class of the delay difference systems

x(t)=

p j=1

Ajx(t− rj), (3)

where the Aj are n-by-n real matrices and the rj are positive real numbers. This case

corresponds to have in (1), ν as a step function of the form

ν(θ )=

p j=1

H (θ− θj)Aj, (4)

where H denotes the Heaviside function,−1 < θ1<· · · < θp< 0, and r(θ ) is any

contin-uous and positive function on[−1, 0] such that r(θj)= rj, for j= 1, . . ., p.

The oscillatory behavior of this class of systems is studied in [7]. A specific treatment for discrete difference systems is included in [3].

As is well known the systems (1), (2), and (3) can be looked, respectively, as particular cases of the differential systems of neutral type

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d dt x(t)− 0 −1 xt− r(θ)dν(θ ) = 0 −1 xt− r(θ)dη(θ ), d dt x(t)− 0 −r x(t+ θ) dν(θ ) = 0 −r x(t+ θ) dη(θ ), (5) d dt x(t)− p j=1 Ajx(t− rj) = p j=1 Bjx(t− rj). (6)

Several criteria for having (6) oscillatory can be found in [2–4], but in all of them, the ma-trices Bjcannot be null, which excludes necessarily the system (3). In [8], the Theorem 3.3

and the Corollaries 4.2 and 4.3, are oscillation criteria obtained in the regard of the systems (5) and (6), which can as well include the systems (2) and (3), respectively. However the results which will be presented here are of different kind.

According to [9], the analysis of the oscillatory behavior of solutions of the system (1), can be based upon the existence or absence of real zeros of the characteristic equation

det I− 0 −1 exp−λr(θ)dν(θ ) = 0, (7)

where by I we mean the n-by-n identity matrix. In fact, in this framework, one can con-clude that (1) is oscillatory if and only if (7) has no real roots, that is, if and only if

1 /∈ σ 0 −1 exp−λr(θ)dν(θ ) (8)

for every λ∈ R; nonoscillatory solutions will exist, whenever (7) has at least a real root. We will denote by BVn the space of all functions of bounded variation, η :[−1, 0] → Mn(R). The space BV1, of all real functions of bounded variation on [−1, 0], will be

denoted simply by BV. For φ∈ BV by

0 −1

dφ(θ ),

we will mean the total variation of φ on [−1, 0]. We notice that if η ∈ BVn and with

j, k= 1, . . ., n, η(θ) = [ηj k(θ )], then each function ηj k∈ BV. The matrix |η| = 0 −1 dηj k(θ ) .

will also be considered.

For any η∈ BVn we can formulate the right and left hand limit matrices at any point

θ∈ [−1, 0], η(θ+) and η(θ−), as well as the right and left hand oscillation matrices Ω_η+(θ )= η(θ+)− η(θ) and Ω_η−(θ )= η(θ) − η(θ−).

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Nonnegative matrices enable us to consider monotonic n-by-n matrix valued functions. For this purpose we recall that a n-by-n real matrix C= [cj k] (j, k = 1, . . ., n) is said

to be nonnegative (positive) whenever cj k 0, (respectively, cj k> 0) for every j, k=

1, . . . , n. These properties will be expressed as usual, through the notations C 0 and

C > 0, respectively. More generally given two n-by-n real matrices, C and D, we will say

that C D (C < D) if D − C 0 (respectively, D − C > 0).

Therefore we will say that a function η :[−1, 0] → Mn(R) is nondecreasing

(nonin-creasing) on a interval J ⊂ [−1, 0], if for every θ1, θ2∈ J such that θ1< θ2 one has

η(θ1) η(θ2) (respectively, η(θ2) η(θ1)); η will be said increasing (decreasing) on J ,

if η is nondecreasing (respectively, nonincreasing) on J and there exist θ1, θ2∈ J such

that θ1< θ2 and η(θ1) < η(θ2) (respectively, η(θ2) < η(θ1)). If for every ε > 0,

suffi-ciently small, η is increasing (decreasing) in[θ − ε, θ + ε] ([−ε, 0] if θ = 0, [−1, −1 + ε] if θ = −1) we will say that θ is a point of increase of η (respectively, a point of decrease of η).

As is well known, any function φ∈ BV can be decomposed as the difference of two nondecreasing functions α and β: φ= α − β. This decomposition is not unique and a particular decomposition of φ is given by

φ= ϕ − ψ, (9)

where by ϕ and ψ we denote, respectively, the positive and negative variation of φ, which are defined as follows. For each θ∈ [−1, 0], let Pθ be the set of all partitions P= {−1 =

θ0, θ1, . . . , θk= θ} of the interval [−1, θ] and to each P ∈ Pθassociate the sets

A(P )= j : φ(θj)− φ(θj−1) > 0

and B(P )= j : φ(θj)− φ(θj−1) < 0

.

Then ϕ and ψ are defined as

ϕ(θ )= sup j∈A(P ) φ(θj)− φ(θj−1) : P∈ Pθ , ψ(θ )= sup j∈B(P ) φ(θj)− φ(θj−1): P∈ Pθ

(whenever A(P ) or B(P ) are empty, we make ϕ(θ )= 0, ψ(θ) = 0). One easily sees that both ϕ and ψ are nondecreasing functions such that φ(θ )= ϕ(θ) − ψ(θ), for every θ ∈

[−1, 0].

These facts can be extended for functions η∈ BVn. In fact, since for each θ∈ [−1, 0]

we have η(θ )= [ηj k(θ )] (j, k = 1, . . ., n) where ηj k ∈ BV, for every j, k = 1, . . ., n,

then decomposing each function ηj k (j, k= 1, . . ., n) as the difference of two

nonde-creasing functions αj k and βj k, the n-by-n matrix valued functions A(θ )= [αj k(θ )],

B(θ )= [βj k(θ )], are both nondecreasing functions in BVn, and for every θ∈ [−1, 0], we

have

η(θ )= A(θ) − B(θ). (10)

If for every j, k= 1, . . ., n, we decompose each function ηj k according to (9) then we

obtain

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with Φ(θ )= [ϕj k(θ )] and Ψ (θ) = [ψj k(θ )], where ϕj k and ψj kare, respectively, the

pos-itive and negative variation of ηj k (j, k= 1, . . ., n).

2. Nonoscillatory solutions

Nonnegative matrices can play some role on the study of the oscillatory behavior of the system (1).

According to the Perron–Frobenius theorem, a nonnegative matrix C∈ Mn(R) has

sev-eral important spectral properties (see [10,11]). As a matter of fact, denoting by σ (C) the spectrum of C and by ρ(C) the spectral radius of C, one has that ρ(C)∈ σ(C). Moreover,

ρ(C) > 0 if C > 0 and 0 C D ⇒ ρ(C) ρ(D).

For a matrix C∈ Mn(R), considering the upper bound

s(C)= max Re z: z∈ σ(C),

of the set Re σ (C)= {Re z: z ∈ σ(C)}, then s(C) = ρ(C) ∈ σ(C) whenever C 0. Through that same theorem, one can conclude that s(C)∈ σ(C), if C = [cj k] (j, k =

1, . . . , n), is essentially nonnegative—that is, if the off-diagonal entries of C (cj kfor j= k)

are nonnegative real numbers. Therefore, if the matrix

0 −1

exp−λr(θ)dν(θ ) (12)

is essentially nonnegative, for every real λ, the spectral set

σ

0

−1

exp−λr(θ)dν(θ )

is dominated by the value

s(λ)= s 0 −1 exp−λr(θ)dν(θ ) , that is, s(λ)∈ σ 0 −1 exp−λr(θ)dν(θ ) ∀λ ∈ R. (13)

On this purpose, we notice that for every real λ, the matrix (12) is nonnegative when the function ν is nondecreasing on [−1, 0], and is essentially nonnegative when for ν(θ) =

[νj k(θ )] (j, k = 1, . . ., n), the off-diagonal functions νj k(θ ) (j= k) are nondecreasing on [−1, 0]. Moreover the assumption (13) is also fulfilled when for each θ ∈ [−1, 0], ν(θ) is

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Under (13), the continuous dependence of the spectrum with respect to parameters en-able us to handle condition (8) in a more suiten-able manner.

Theorem 1. Under assumption (13), the system (1) is oscillatory if and only if s(λ) < 1,

for every real λ.

Proof. Noticing that

s(λ) 0 −1 exp−λr(θ)dν(θ ) ,

we claim that limλ→+∞s(λ)= 0.

In fact, if r(θ ) is a positive function on[−1, 0], then letting

m(r)= min r(θ ): −1 θ 0,

one has for every real λ 0

0 −1 exp−λr(θ)dν(θ ) exp −λm(r) 0 −1 dν(θ ), and so s(λ)→ 0 as λ → +∞.

If r(θ ) is a positive function on[−1, 0[ and r(0) = 0, as then ν(θ) is atomic at zero, for every ε > 0 there exists a real γ > 0 such that

0 −γ

dν(θ )<ε

2.

Thus taking m0= min{r(θ): −1 θ −γ } > 0, we have for every real λ 0 0 −1 exp−λr(θ)dν(θ ) −γ −1 exp−λr(θ)dν(θ ) + 0 −γ exp−λr(θ)dν(θ ) exp(−λm0) −γ −1 dν(θ ) + 0 −γ dν(θ ),

and by consequence, for λ arbitrarily large, we conclude that

0 −1 exp−λr(θ)dν(θ ) < ε.

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Hence, also in this case, s(λ)→ 0 as λ → +∞.

Therefore supposing that (1) is oscillatory and that for some real λ0 it is s(λ0) 1,

by continuity, there exists a real λ1 λ0 such that s(λ1)= 1, which, in view of (13),

contradicts (8). ✷

Remark 2. The proof of this theorem enable us to conclude that under the assumption (13),

the system (1) has at least a nonoscillatory solution whenever s(λ)→ +∞, as λ → −∞. Taking a decomposition of ν∈ BVnaccording to (10),

ν= A − B,

where A, B ∈ BVn are nondecreasing functions, the matrix (12) is decomposed into the

difference 0 −1 exp−λr(θ)dν(θ )= 0 −1 exp−λr(θ)dA(θ )− 0 −1 exp−λr(θ)dB(θ ).

The following theorem states the existence of, at least, a nonoscillatory solution.

Theorem 3. Let θ0∈ [−1, 0] be such that r(θ0)= r . If either

Ω_A+(θ0) >|B| or Ω_A−(θ0) >|B|

then (1) has at least a nonoscillatory solution.

Proof. Let us assume, for example, that Ω_A+(θ0) >|B|.

For ε > 0 small enough, we have

0 −1 exp−λr(θ)dA(θ ) θ0+ε θ0 exp−λr(θ)dA(θ ).

By application of a mean value property of the functions of bounded variation, we can obtain for every real λ < 0,

θ0+ε

θ0

exp−λr(θ)dA(θ ) exp−λr(θ0+ δε)

A(θ0+ ε) − A(θ0)

,

for some δ∈ ]0, 1[, depending upon r, θ0, ε and A. Then, making ε→ 0+, we have 0

−1

exp−λr(θ)dA(θ ) exp−λ r Ω_A+(θ0).

On the other hand, for every real λ < 0, the following matrix inequality holds

0

0 −1

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and by consequence − 0 −1 exp−λr(θ)dB(θ ) − exp−λ r |B|.

Therefore, for every real λ < 0, we have that

0 −1 exp−λr(θ)dν(θ ) exp−λ r Ω_A+(θ0)− |B| . (14)

As the matrix Ω_A+(θ0)− |B| is positive, this means in particular that also 0

−1

exp−λr(θ)dν(θ )> 0,

for every λ < 0. Thus not only assumption (13) is fulfilled but also

s(λ)= ρ 0 −1 exp−λr(θ)dν(θ ) .

But from (14) we can conclude that

ρ 0 −1 exp−λr(θ)dν(θ ) exp−λ r ρΩ_A+(θ0)− |B|

and by consequence s(λ)→ +∞, as λ → −∞. Hence by the Remark 2, (1) has at least a nonoscillatory solution. ✷

The behavior of the function ν at the point θ0∈ [−1, 0] where is attained the absolute

maximum of the delay function r(θ ), has a specific relevance, as was already shown in [12] for the scalar case of (1).

Theorem 4. Let θ0∈ [−1, 0] be such that r(θ0)= r and r(θ) < r for every θ = θ0. If

θ0is a point of increase of ν, then (1) has at least a nonoscillatory solution.

Proof. For a matter of simplicity, let us assume that θ0= −1.

Considering the decomposition of ν, given by (11), then for some ε > 0, the matrix

Ψ (θ ) is constant on[−1, −1 + ε] and by consequence, on this interval

Φ(θ )= ν(θ) − C,

for some real n-by-n real matrix C. Therefore

0 −1

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= 0 −1 exp−λr(θ)dΦ(θ )− 0 −1+ε exp−λr(θ)dΨ (θ ) −1+ε −1 exp−λr(θ)dΦ(θ )− 0 −1+ε exp−λr(θ)dΨ (θ ).

Take 0 < δ < ε such that

m= min r(θ ): θ∈ [−1, −1 + δ]> M= max r(θ ): θ∈ [−1 + ε, 0].

One easily can see that the following matrix relations hold, for every real λ < 0:

0 0 −1+ε exp−λr(θ)dΨ (θ ) exp(−λM)|Ψ |, −1+ε −1 exp−λr(θ)dΦ(θ ) exp(−λm)Φ(−1 + ε) − Φ(−1).

Thus we obtain for every real λ < 0,

−1+ε −1 exp−λr(θ)dΦ(θ ) exp(−λm)ν(−1 + ε) − ν(−1), − 0 −1+ε exp−λr(θ)dΨ (θ ) − exp(−λM)|Ψ |,

which imply that

0 −1

exp−λr(θ)dν(θ )

exp(−λm)ν(−1 + ε) − ν(−1)− expλ(m− M)|Ψ |. (15) Since the nonnegative matrix exp(λ(m− M))|Ψ | tends to the null matrix, as λ → −∞, we can conclude that there exists a real number 1 > 0 sufficiently large, such that, for every

λ <−1, (ν(−1 + ε) − ν(−1)) − exp(λ(m − M))|Ψ | is a positive matrix. Thus, for every λ <−1, we have, in particular,

0 −1

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and so assumption (13) is satisfied in a way that s(λ)= ρ 0 −1 exp−λr(θ)dν(θ ) .

Moreover (15) implies that, for every λ <−1,

ρ 0 −1 exp−λr(θ)dν(θ ) exp(−λm)ρν(−1 + ε) − ν(−1)− expλ(m− M))|Ψ |.

Since the right hand member of this inequality tends to+∞, as λ → −∞, we can state, as in Theorem 1, that (1) has at least a nonoscillatory solution. ✷

Theorem 4 can be applied to the system (3) and the following corollary can easily be obtained.

Corollary 5. If rk= max{rj: j= 1, . . ., p} and Ak> 0 then (3) has at least a

nonoscilla-tory solution.

Proof. As a matter of fact, if Ak> 0 then θkis a point of increase of ν(θ )= p

j=1H (θ−

θj)Aj. Therefore choosing r(θ ) continuous and positive on[−1, 0] in manner that r(θk)= r and r(θ) < r for every θ = θk, one can conclude by Theorem 4 that (3) has at least

a nonoscillatory solution. ✷

Similar arguments enable us to state the following theorem.

Theorem 6. If ν is increasing on[−1, 0] and nonconstant on [−1, 0[ then (1) has at least

a nonoscillatory solution.

Proof. Since ν is increasing on[−1, 0[, there exist θ1, θ2∈ [−1, 0[ such that θ1< θ2and

2= ν(θ2)− ν(θ1)

is a positive matrix. Then the following matrix relation holds for every real λ 0:

0 −1 exp−λr(θ)dν(θ ) θ2 θ1 exp−λr(θ)dν(θ ) exp(−λm)2, where m= min r(θ ): θ∈ [θ1, θ2] > 0.

Thus as before, assumption (13) is fulfilled, s(λ)→ +∞, as λ → −∞, and (1) has at least a nonoscillatory solution. ✷

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3. Explicit conditions for oscillations

As is well known matrix measures are a relevant tool on the oscillation theory of delay systems. For a matter of completeness we recall briefly, its definition and the properties which will be used in the sequel.

For each induced norm, · , in Mn(R), we associate a matrix measure µ : Mn(R) → R,

which is defined for any C∈ Mn(R) as

µ(C)= lim

γ→0+

I + γ C − 1

γ ,

where by I we mean the identity matrix.

Well known matrix measures of a matrix C=cj k ∈ Mn(R), are µ1(C)= max ckk+ j=k |cj k|: k = 1, . . ., n , µ_∞(C)= max cjj+ k=j |cj k|: j = 1, . . ., n ,

which correspond, respectively, to the induced norms inMn(R) given by: C 1= max _n j=1 |cj k|: k = 1, . . ., n , C ∞= max _n k=1 |cj k|: j = 1, . . ., n .

Independently of the considered induced norm inMn(R), a matrix measure has always

the following properties (see [13]): (i) s(C) µ(C) C .

(ii) µ(C1)− µ(−C2) µ(C1+ C2) µ(C1)+ µ(C2) (C1, C2∈ Mn(R)).

(iii) µ(γ C)= γ µ(C), for every γ 0.

If η∈ BVn, the continuity of µ onMn(R) implies that µ ◦ η ∈ BV; in consequence, with [a, b] ⊂ [−1, 0], the following inequalities hold (see [8]):

(iv) If φ∈ C([a, b]; R) is nonincreasing and positive, then

µ b a φ(θ ) dη(θ ) b a φ(θ )dµη(θ )− η(a).

(v) If φ∈ C([a, b]; R) is nondecreasing and positive, then

µ b a φ(θ ) dη(θ ) − b a φ(θ ) dµη(b)− η(θ).

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By (i), if for some matrix measure µ, µ 0 −1 exp−λr(θ)dν(θ ) < 1 ∀λ ∈ R, (16)

we can conclude that system (1) is oscillatory.

For any η∈ BVn, the functions η0and η1of BVn, given, respectively, by

η0(θ )= η(0) − η(θ), η1(θ )= η(θ) − η(−1) (θ ∈ [−1, 0]),

will be considered in the sequel.

In the following theorem we obtain conditions for having (1) oscillatory with respect to the family of all monotonic delay functions.

Theorem 7. If on[−1, 0], µ ◦ ν1is nonincreasing and µ◦ ν0is nondecreasing then (1) is

oscillatory for all monotonic delay functions r(θ ).

Proof. Assume that r: [−1, 0] → R is a monotonic function.

By (iv) and (v), if r(θ ) is nonincreasing we have that

µ 0 −1 exp−λr(θ)dν(θ ) 0 −1 exp−λr(θ)d(µ◦ ν1)(θ ), if λ 0, µ 0 −1 exp−λr(θ)dν(θ ) − 0 −1 exp−λr(θ)d(µ◦ ν0)(θ ), if λ 0, (17) and if r(θ ) is nondecreasing, µ 0 −1 exp−λr(θ)dν(θ ) 0 −1 exp−λr(θ)d(µ◦ ν1)(θ ), if λ 0, µ 0 −1 exp−λr(θ)dν(θ ) − 0 −1 exp−λr(θ)d(µ◦ ν0)(θ ), if λ 0. (18)

Then take, respectively, for λ∈ R, the functions

g(λ)=      0 −1exp(−λr(θ)) d(µ ◦ ν1)(θ ), if λ < 0, µ(2ν), if λ= 0, −0 −1exp(−λr(θ)) d(µ ◦ ν0)(θ ), if λ > 0, h(λ)=      −0 −1exp(−λr(θ)) d(µ ◦ ν0)(θ ), if λ < 0, µ(2ν), if λ= 0, 0 −1exp(−λr(θ)) d(µ ◦ ν1)(θ ), if λ > 0,

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where 2ν= ν(0) − ν(−1). Noticing that g(0+)= h(0−)= − 0 −1 d(µ◦ ν0)(θ )= µ ν0(−1) = µν(0)− ν(−1) and g(0−)= h(0+)= 0 −1 d(µ◦ ν1)(θ )= µ ν1(0) = µν(0)− ν(−1),

one can conclude that g and h are both continuous inR.

Since on[−1, 0], µ ◦ ν1is nonincreasing and µ◦ ν0is nondecreasing, one has

g(λ) 0 ∀λ ∈ R,

if r(θ ) is nonincreasing and

h(λ) 0 ∀λ ∈ R,

if r(θ ) is nondecreasing. Hence, for r(θ ) monotonic, by (17) and (18) one has (16) satisfied, which achieves the proof. ✷

Remark 8. The assumptions in the Theorem 7 of µ◦ ν1be nonincreasing and µ◦ ν0be

nondecreasing are fulfilled, if one assumes as in [14], that for θ1, θ2∈ [−1, 0],

θ1< θ2 ⇒ µ

ν(θ2)− ν(θ1)

0. (19)

As a matter of fact, if θ1< θ2, one has by (ii),

(µ◦ ν1)(θ2)− (µ ◦ ν1)(θ1) = µν(θ2)− ν(−1) − µν(θ1)− ν(−1) µν(θ2)− ν(−1) − ν(θ1)+ ν(−1) = µν(θ2)− ν(θ1) 0, and (µ◦ ν0)(θ1)− (µ ◦ ν0)(θ2) = µν(0)− ν(θ1) − µν(0)− ν(θ2) µν(0)− ν(θ1)− ν(0) + ν(θ2) = µν(θ2)− ν(θ1) 0.

Following [8] let us define

α(A1)= µ(A1), α(Aj)= µ _j k=1 Ak − µ _j₋₁ k=1 Ak for j= 2, . . ., p,

β(Ap)= µ(Ap), β(Aj)= µ _p k=j Ak − µ _p k=j+1 Ak for j= 1, . . ., p − 1.

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Therefore, with respect to the system (3), from the Theorem 7 we can state the following corollary.

Corollary 9. If for j= 1, . . ., p, α(Aj) 0 and β(Aj) 0, then system (3) is oscillatory

for every family of delays (r1, . . . , rp)∈ Rp₊.

In particular, as by (ii)

α(Aj) µ(Aj), β(Aj) µ(Aj),

we obtain as in [7]:

Corollary 10. If µ(Ak) 0, for every k = 1, . . ., p, then (3) is oscillatory for every family

of delays (r1, . . . , rp)∈ Rp₊.

In the following we will analyze what happens when r(θ ) is not a monotonic function on[−1, 0].

For that purpose we notice that by property (ii) of the matrix measures,

µ 0 −1 exp−λr(θ)dν(θ ) µ θ0−δ −1 exp−λr(θ)dν(θ ) + µ θ0+δ θ0−δ exp−λr(θ)dν(θ ) + µ 0 θ0+δ exp−λr(θ)dν(θ ) , (20)

for every θ0∈ [−1, 0] and δ 0, small enough.

Theorem 11. Let r(θ ) be differentiable and positive on[−1, 0], increasing on [−1, θ0−δ],

constant on[θ0− δ, θ0+ δ] and decreasing on [θ0+ δ, 0]. If

µ(ν(θ0+ δ) − ν(θ0− δ)) 0, (21) µν(θ0− δ) − ν(θ) 0, (µ ◦ ν1)(θ ) 0, for every θ ∈ [−1, θ0− δ], (22) µν(θ )− ν(θ0+ δ) 0, (µ ◦ ν0)(θ ) 0, for every θ ∈ [θ0+ δ, 0], (23) and θ0−δ −1 (µ◦ ν1)(θ ) d log r(θ )− 0 θ0+δ (µ◦ ν0)(θ ) d log r(θ )< e, (24) then (1) is oscillatory.

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Proof. We will prove that (16) holds.

For λ= 0, by (20), (21) and the first part of (22) and (23) we can conclude that

µν(0)− ν(−1) µν(0)− ν(θ0+ δ) + µν(θ0+ δ) − ν(θ0− δ) + µν(θ0− δ) − ν(−1) 0.

Let now λ < 0. By (20) and the properties (iv) and (v) of the matrix measures we obtain

µ 0 −1 exp−λr(θ)dν(θ ) − θ0−δ −1 exp−λr(θ)dµν(θ0− δ) − ν(θ) + exp−λr(θ0) µν(θ0+ δ) − ν(θ0− δ) + 0 θ0+δ exp−λr(θ)dµν(θ )− ν(θ0+ δ) .

Therefore, integrating by parts, we have

µ 0 −1 exp−λr(θ)dν(θ ) exp−λr(−1)µν(θ0− δ) − ν(−1) − θ0−δ −1 λ exp−λr(θ)µν(θ0− δ) − ν(θ) dr(θ ) + exp−λr(θ0) µν(θ0+ δ) − ν(θ0− δ) + exp−λr(0)µν(0)− ν(θ0+ δ) + 0 θ0+δ λ exp−λr(θ)µν(θ )− ν(θ0+ δ) dr(θ ).

Then by (21) and the first part of (22) and (23) we state that, for every λ < 0,

µ 0 −1 e−λr(θ)dν(θ ) 0.

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µ 0 −1 exp−λr(θ)dν(θ ) exp−λr(θ0− δ) µν(θ0− δ) − ν(−1) + θ0−δ −1 λ exp−λr(θ)(µ◦ ν1)(θ ) dr(θ ) + exp−λr(θ0) µν(θ0+ δ) − ν(θ0− δ) + exp−λr(θ0+ δ) µν(0)− ν(θ0+ δ) − 0 θ0+δ λ exp−λr(θ)(µ◦ ν0)(θ ) dr(θ ) θ0−δ −1 λ exp−λr(θ)(µ◦ ν1)(θ ) dr(θ )− 0 θ0+δ λ exp−λr(θ)(µ◦ ν0)(θ ) dr(θ ).

Noticing that the function u exp(−u) has an absolute maximum at u = 1, we obtain for every θ∈ [−1, 0]

λ exp−λr(θ) e

−1

r(θ ).

As r(θ ) is increasing in[−1, θ0−δ[ and decreasing in ]θ0+δ, 0], by the second part of (22)

and (23) we obtain µ 0 −1 exp−λr(θ)dν(θ ) 1 e θ_0−δ −1 (µ◦ ν1)(θ ) r(θ ) dr(θ )− 0 θ0+δ (µ◦ ν0)(θ ) r(θ ) dr(θ ) , and (24) implies µ 0 −1 exp−λr(θ)dν(θ ) < 1, for every λ > 0.

This achieves the proof. ✷

Example 12. Let us consider the system (1) for

r(θ )=      −θ2₋6 5θ , if − 1 θ < − 3 5, 9 25, if − 3 5 θ − 2 5, −θ2₋4 5θ+ 5 25, if − 2 5< θ 0,

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and ν(θ )= θθ+2₅ 3 4 −θ+3₅(θ+ 1) .

With respect to the matrix measure µ_∞, (21) is satisfied since

µ_∞ ν −2 5 − ν −3 5 = µ∞ −3 25 0 0 −₂₅3 < 0.

The same holds to assumptions (22) and (23), since for every θ∈ [−1, −3/5],

µ_∞ ν −3 5 − ν(θ) = µ∞ ₃ 25− θ θ+2₅ 0 0 θ+3₅(θ+ 1) = θ+3 5 (θ+ 1) 0, µ_∞ν(θ )− ν(−1)= µ_∞ θθ+2₅−3₅ 0 0 −θ+3₅(θ+ 1) = θ θ+2 5 −3 5 0, and, for every θ∈ [−2/5, 0],

µ_∞ ν(θ )− ν −2 5 = µ∞ θθ+2₅ 0 0 −θ+3₅(θ+ 1) +₂₅3 = θ θ+2 5 0, µ_∞ν(0)− ν(θ)= µ_∞ −θθ+2₅ 0 0 −3₅+θ+3₅(θ+ 1) = −3 5+ θ+3 5 (θ+ 1) 0. Moreover, as −3/5 −1 µ(ν(θ )− ν(−1)) r(θ ) dr(θ )− 0 −2/5 µ(ν(0)− ν(θ)) r(θ ) dr(θ ) = − −3/5 −1 θ+3₅(θ+ 1)2θ+6₅ θθ+6₅ dθ+ 0 −2/5 θθ+2₅2θ+4₅ (θ+ 1)θ−1₅ dθ = − 8 25− 36 25ln 3+ 6 5ln 5 < e,

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Making in the Theorem 11, δ= 0, we obtain the following corollary.

Corollary 13. Let r :[−1, 0] → R+ be differentiable on[−1, 0], increasing on [−1, θ0]

and decreasing on[θ0, 0]. If for every θ ∈ [−1, θ0],

µν(θ0)− ν(θ) 0, (µ◦ ν1)(θ ) 0, (25) for θ∈ [θ0, 0] µν(θ )− ν(θ0) 0, (µ◦ ν0)(θ ) 0, (26) and θ0 −1 (µ◦ ν1)(θ ) d log r(θ )− 0 θ0 (µ◦ ν0)(θ ) d log r(θ )< e, (27) then (1) is oscillatory.

This corollary is illustrated in the following example.

Example 14. Consider (1) with r(θ )= −θ(θ + 1) and

ν(θ )= θ (θ+ 1)θ+1₂ 5 0 −6θ − 7 .

For the matrix measure µ1, we have for every θ∈ [−1, −1/2]

µ1 ν −1 2 − ν(θ) = µ1 −θ(θ + 1)θ+1₂ 0 0 3+ 6θ = −θ(θ + 1) θ+1 2 0, µ1(ν(θ )− ν(−1)) = µ1 θ (θ+ 1)θ+1₂ 0 0 −6θ − 6 = θ(θ + 1) θ+1 2 0,

and for every θ∈ [−1/2, 0],

µ1 ν(θ )− ν −1 2 = µ1 θ (θ+ 1)θ+1₂ 0 0 −3 − 6θ = θ(θ + 1) θ+1 2 0, µ1 ν(0)− ν(θ)= µ1 −θ(θ + 1)θ+1₂ 0 0 6θ = −θ(θ + 1) θ+1 2 0.

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Therefore conditions (25) and (26) of the Corollary 13 are fulfilled. The same holds to assumption (27), since −1/2 −1 θ (θ+ 1)θ+1₂ −θ(θ + 1) d −θ(θ + 1)− 0 −1/2 −θ(θ + 1)θ+1₂ −θ(θ + 1) d −θ(θ + 1) = −1/2 −1 θ+1 2 (2θ+ 1) dθ + 0 −1/2 θ+1 2 (2θ+ 1) dθ = 1 6< e. Hence the corresponding system (1) is oscillatory.

Now, assuming that δ= 0, as before, and further that θ0= −1, the following corollary

can be stated.

Corollary 15. Let on[−1, 0], be (µ ◦ ν0)(θ ) 0, (µ ◦ ν1)(θ ) 0 and r(θ) positive,

differ-entiable and decreasing. Then system (1) is oscillatory if

0 −1 (µ◦ ν0)(θ ) d log r(θ )>−e. (28)

Proof. Just notice that, in this case, (25) is fulfilled, since µ(0)= 0. On the other hand,

assumption (26) becomes equivalent to (µ◦ ν0)(θ ) 0 and (µ ◦ ν1)(θ ) 0, for every

θ∈ [−1, 0], while (27) gives rise to (28). ✷

Analogously, for δ= 0 and θ0= 0, we obtain the following corollary.

Corollary 16. Let on[−1, 0], be (µ ◦ ν0)(θ ) 0, (µ ◦ ν1)(θ ) 0 and r(θ) positive,

differ-entiable and increasing. Then system (1) is oscillatory if

0 −1 (µ◦ ν1)(θ ) d log r(θ )< e.

The following example illustrates the Corollary 15.

Example 17. Let us consider (1) for

ν(θ )= θ2 −θ θ −2θ and r(θ )= 1 − θ.

We have, for the matrix measure µ_∞,

(µ_∞◦ ν0)(θ )= µ∞ −θ2 _θ −θ 2θ = max{−θ2_{− θ, θ} = −θ(θ + 1) 0,}

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for every θ∈ [−1, 0], and µ_∞◦ ν1(θ ) = µ∞ θ2− 1 −θ − 1 θ+ 1 −2θ − 2 = max{θ2_{+ θ, −θ − 1} = θ(θ + 1) 0,} for every θ∈ [−1, 0]. As 0 −1 θ (θ+ 1) 1− θ dθ= 0 −1 −θ − 2 + 2 1− θ dθ= −θ2 2 − 2θ − ln(1 − θ) 2 0 −1 =1 2 − 2 + ln 4 ≈ −0, 2 > −e, the corresponding system (1) is then oscillatory.

Considering in (1), ν(θ ) given by (4), for−1 < θ1<· · · < θp< 0, and r(θ )

differen-tiable, decreasing and positive on[−1, 0], one obtains the system (3) with rj = r(θj) for

j = 1, . . ., p, such that r1>· · · > rp. In this situation we can apply the Corollary 15 and

consequently obtain the following, corollary.

Corollary 18. Let µ _p k=j Ak 0 and µ _j k=1 Ak 0, (29)

for every j= 1, . . ., p. Then system (3) is oscillatory if

p j=2 µ _p k=j Ak log rj rj−1 >−e. (30)

Proof. The conditions corresponding to (µ◦ ν0)(θ ) 0 and (µ ◦ ν1)(θ ) 0 in the

Corol-lary 15, are in this case, respectively, µ(p_k_=jAk) 0 and µ( j

k=1Ak) 0, for every

j = 1, . . ., p.

On the other hand, as then µ(p_k₌₁Ak)= 0, (28) becomes,

0 −1 (µ◦ ν0)(θ ) d log r(θ ) = θ2 θ1 µ _p k=2 Ak dlog r(θ )+ · · · + θp θp−1 µ(Ap) d log r(θ ) + 0 θp µ(0) dlog r(θ )

(21)

= µ _p k₌₂ Ak logr2 r1 + · · · + µ _p k_=p−1 Ak logrp−1 rp−2 + µ(A p) log rp rp−1 = p j=2 µ _p k=j Ak log rj rj−1 >−e,

which proves the corollary. ✷

An analogous result can be obtained for (3) through the Corollary 16, by considering the system (1) with ν(θ ) given by (4), for−1 < θ1<· · · < θp< 0, and r(θ ) differentiable

and increasing on [−1, 0]. In fact, now one obtains the system (3) with rj = r(θj) for

j = 1, . . ., p, such that r1<· · · < rp. Hence the following corollary can be stated.

Corollary 19. Let µ _p k=j Ak 0 and µ _j k=1 Ak 0,

for every j= 1, . . ., p. Then system (3) is oscillatory if

p−1 j=1 µ _j k=1 Ak logrj+1 rj < e.

The following example illustrates the Corollary 18.

Example 20. Let us consider the system

x(t)= A1x t−7 4 + A2x t−3 2 + A3x t−5 4 , (31) where A1= −3 −5 1 −9 , A2= 1 2 −2 5 , A3= −1 1 4 −3 . We have µ1(A1)= max{−2, −4} = −2 0, µ1(A1+ A2)= µ1 −2 −3 −1 −4 = max{−1, −1} = −1 0, µ1(A1+ A2+ A3)= µ1 −3 −2 3 −7 = max{0, −5} = 0, µ1(A2+ A3)= µ1 0 3 2 2 = max{2, 5} = 5 0, µ1(A3)= max{3, −2} = 3 0.

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µ ₃ k=2 Ak lnr2 r1+ µ ₃ k=3 Ak lnr3 r2 = 5 ln3/2 7/4+ 3 ln 5/4 3/2= 5 ln 6 7+ 3 ln 5 6≈ −1, 32 > −e. Hence the system (31) is oscillatory.

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